What is a Topology?
A topology on a set X is a collection τ (tau) of subsets of X, called open sets, that satisfies three fundamental axioms. The pair (X, τ) is called a topological space.
Unlike metric spaces, topology doesn't care about distances — only about which sets are declared "open." This makes topology extraordinarily general: the same framework describes the real line, a donut, a sphere, and stranger objects alike.
The Basic Setup
Let X be any non-empty set — the "universe" of points. A topology τ on X is a family of subsets satisfying the three axioms on the next tab. We call the elements of τ the open sets of the topology.
A set X with a collection of subsets declared as open. The open sets must include ∅ and X, and be closed under unions and finite intersections.
The Three Axioms
Both the empty set and the entire space must be open sets in every topology.
The union of any collection of open sets — even infinitely many — must also be an open set.
then ⋃ Uᵢ ∈ τ
The intersection of any finite collection of open sets must also be open. (Infinite intersections are not required to be open.)
then U₁ ∩ ... ∩ Uₙ ∈ τ
Why finite intersections only?
Infinite intersections of open intervals can collapse to a single point, which is closed — not open. For example, in ℝ with the standard topology:
The intersection of infinitely many open intervals narrows down to the single point {0}, which is not open in the standard topology on ℝ. This is why Axiom 3 requires only finite intersections.
Visual demonstration of Axioms 2 and 3: two open sets U and V (left), their union U∪V (middle, must be open), and their intersection U∩V (right, must be open for finite collections).
Simple Topologies
Let X = {a, b, c}. How many distinct topologies can we define on X? (The answer is 29.) Here are the most illuminating ones:
| Name | Open Sets τ | Complexity | Notes |
|---|---|---|---|
| Indiscrete | {∅, X} | simple | Minimal. No points can be separated. |
| Discrete | {∅,{a},{b},{c},{a,b},{a,c},{b,c},X} | simple | Maximal. Every subset is open. |
| Particular Point | {∅, {a}, {a,b}, {a,c}, X} | medium | Every open set must contain point a. |
| Excluded Point | {∅, {b}, {c}, {b,c}, X} | medium | Every proper open set excludes point a. |
| Sierpiński Space | {∅, {1}, {0,1}} | simple | X = {0,1}. The simplest non-trivial T₀ space. |
Only ∅ and X are open. Points are completely inseparable — no open set can distinguish between them.
Every subset is open. Points are maximally separated — every singleton {a}, {b}, {c} is its own open set.
Every non-empty open set must contain point a. It acts as a "mandatory member" of all open neighborhoods.
X = {0,1}. The open sets are ∅, {1}, and {0,1}. The simplest space that is T₀ but not T₁.
Complex Examples
Standard Topology on ℝ
The most important topology in analysis. A subset U ⊆ ℝ is open if every point in U has an open interval around it entirely contained in U. Open sets are arbitrary unions of open intervals (a,b).
Open sets in ℝ are unions of open intervals. Endpoints are excluded. Any union of such sets is also open.
A set is open iff its complement is finite (or it's ∅). This is coarser than the standard topology.
Product Topology
Given two topological spaces (X, τ_X) and (Y, τ_Y), the product topology on X × Y has as basis all sets of the form U × V where U is open in X and V is open in Y. The plane ℝ² with its standard topology is the product of ℝ with itself.
Open sets in ℝ² are unions of open rectangles (U × V). A disk is also open — as a union of infinitely many open rectangles.
Quotient Topology
Given a space X and an equivalence relation ~, the quotient space X/~ identifies equivalent points. The quotient topology makes the projection map q: X → X/~ continuous. Gluing the ends of an interval produces a circle; gluing opposite edges of a square produces a torus.
Identify endpoints 0 ~ 1 in [0,1]. The quotient space is homeomorphic to S¹.
Identify top↔bottom and left↔right edges of [0,1]². The quotient is the torus T² = S¹ × S¹.
Subspace Topology
Given (X, τ) and a subset A ⊆ X, the subspace topology on A is τ_A = {U ∩ A : U ∈ τ}. The open sets of A are exactly the intersections of open sets of X with A.
The interval (0,1) is open in ℝ. Its intersection with [0,1] is (0,1) — still open in the subspace. But {0} ∩ (−ε, ε) = {0} is open in the subspace {0,1} of ℝ.
Key Concepts
Any two distinct points can be separated by disjoint open sets. Most "reasonable" spaces are Hausdorff.
A space is connected if it cannot be split into two disjoint non-empty open sets. ℝ is connected; {0,1} (discrete) is not.
Every open cover has a finite subcover. Compact sets generalize "closed and bounded" from ℝⁿ to arbitrary spaces.
f: X→Y is continuous if the preimage of every open set in Y is open in X. This replaces the ε-δ definition.
A bijection f: X→Y where both f and f⁻¹ are continuous. Homeomorphic spaces are "topologically identical."
A collection ℬ of open sets such that every open set is a union of basis elements. A basis generates the topology.
Connected vs disconnected spaces. A space is disconnected if it decomposes into two disjoint clopen sets.
[0,1] is compact: any open cover reduces to finitely many sets. (0,1) is not compact: the cover (1/n,1) has no finite subcover.